Monoid Hecke Algebras

This paper concerns the monoid Hecke algebras H introduced by Louis Solomon. We determine explicitly the unities of the orbit algebras associated with the two-sided action of the Weyl group W . We use this to: 1. find a description of the irreducible representations of H, 2. find an explicit isomorphism between H and the monoid algebra of the Renner monoid R, 3. extend the Kazhdan-Lusztig involution and basis to H, and 4. prove, for a W ×W orbit of R, the existence (conjectured by Renner) of generalized Kazhdan-Lusztig polynomials. Introduction A monoid analogue of the Iwahori-Hecke algebra [11] was obtained by Solomon [27]–[29]. In an earlier paper [19] the author studied Solomon’s monoid Hecke algebras by studying the associated orbit algebras. These orbit algebras arise from the two-sided action of the Weyl group W on the Renner monoid R. In particular, the coefficients of the unity of the empty level orbit algebra were shown to be Rx,y, where Rx,y are polynomials introduced by Kazhdan and Lusztig [12]. The other orbit algebras were also shown to have unities, but their coefficients were only implicitly given. In this paper we give an explicit formula for the unities of all the orbit algebras, thereby obtaining a description of the irreducible representations of monoid Hecke algebras. We also obtain an explicit, but very complicated, isomorphism between the monoid Hecke algebra and the monoid algebra of R, solving a problem posed by Solomon [28]. We go on to extend to the monoid Hecke algebra the Kazhdan-Lusztig involution and basis for the (group) Iwahori-Hecke algebra. This then immediately yields polynomials Pθ,σ for θ, σ in the same W ×W orbit of R, partially solving a problem posed by Renner [26]. These polynomials are still mysterious; however, in the simplest case they are products of relative KazhdanLusztig polynomials introduced by Deodhar [6]. 1. Reductive monoids and monoids of Lie type Consider the general linear group G = GLn(F ) over an algebraically closed field F . It is the unit group of the multiplicative monoidM = Mn(F ) of all n×nmatrices over F . This monoid has the following structure. The diagonal idempotents form a Boolean lattice with respect to the natural order of idempotents: f ≤ e if ef = fe = f. Received by the editors December 3, 1993. 1991 Mathematics Subject Classification. Primary 20G40, 20G05, 20M30. Research partially supported by NSF Grant DMS9200077. c ©1997 American Mathematical Society